The hypotenuse of a right triangle whose legs are consecutive whole numbers is 29 units. What is the sum of the lengths of the two legs?
Answer: Let's call the length of the shorter leg $x$. Then the length of the longer leg is $x + 1$. Using the Pythagorean Theorem, we write the equation $x^2 + (x + 1)^2 = 29^2$ and solve for $x$. Expanding $(x + 1)^2$, we get $x
^2 + x^2 + 2x + 1 = 841$. This can be simplified to $2x^2 + 2x = 840$, or $x^2 + x = 420$. Factoring out $x$ on the left, we can rewrite it as $x(x + 1) = 420$. In other words, the product of these two consecutive numbers is 420, which means they must each be close to the square root of 420. Indeed, $20 \times 21 = 420$, so the legs must be 20 and 21. Their sum is $20 + 21 = \boxed{41}$.